Trong mặt phẳng tọa độ Oxy, cho hai vectơ u =(4;1) và v =(1;4). Tìm m để vectơ

* Hướng dẫn giải

Ta có $\left\{\begin{array}{l}\overrightarrow{a}=m.\overrightarrow{u}+\overrightarrow{v}=\left(4m+1;m+4\right)\\ \overrightarrow{b}=\overrightarrow{i}+\overrightarrow{j}=\left(1;1\right)\end{array}\right..$

Yêu cầu bài toán $\iff \cos \left(\overrightarrow{a},\overrightarrow{b}\right)=\cos {45}^0=\frac{\sqrt{2}}{2}$

$\iff \frac{\left(4m+1\right).1+\left(m+4\right).1}{\sqrt{2}\sqrt{{\left(4m+1\right)}^2+{\left(m+4\right)}^2}}=\frac{\sqrt{2}}{2}\iff \frac{5\left(m+1\right)}{\sqrt{2}\sqrt{17m^2+16m+17}}=\frac{\sqrt{2}}{2}$

$\begin{array}{l}\iff 5\left(m+1\right)=\sqrt{17m^2+16m+17}\\ \iff \left\{\begin{array}{l}m+1\ge 0\\ 25m^2+50m+25=17m^2+16m+17\end{array}\right.\iff m=-\frac{1}{4}.\end{array}$

Chọn C.